If i have my child fly with another adult do i need to sign a release form or have her take her birth certficate?

When my kids were young and with family and friends and away from me, I always had a consent form of some kind . Medical info , my info, etc.

I don't think you do.

Call the airline. The airport has no control over that.

That's an excellent question for the airlines. I would recommend contacting the airline and requesting that information.